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A Consensus-Based Algorithm for Non-Convex Multiplayer Games: Nonlinear Oligopoly Games by@oligopoly

by OligopolyJuly 10th, 2024

**Authors:**

(1) Enis Chenchene, Department of Mathematics and Scientific Computing, University of Graz;

(2) Hui Huang, Department of Mathematics and Scientific Computing, University of Graz;

(3) Jinniao Qiu, Department of Mathematics and Statistics, University of Calgary.

2.1 Quantitative Laplace principle

2.2 Global convergence in mean-field law

3 Numerical experiments and 3.1 One-dimensional illustrative example

3.2 Nonlinear oligopoly games with several goods

4 Conclusion, Acknowledgments, Appendix, and References

Figure 2: Studying the dependence of Algorithm 1 with respect to the algorithm’s parameters to solve (3.5).

of good produced, namely

**3.2.1 Experimental setup**

**3.2.2 Results and discussio**

In Figure 3, which shows the results corresponding to the experiment in case 1, as observed for instance in [28], we see that the anisotropic case does indeed show a faster convergence rate especially in the initial iterations and for λ which is of the order of σ 2 . If λ increases, though, we see no significant differences in the convergence behavior of anisotropic or isotropic dynamics, both for the residual and for the variance decrease.

Figure 4 shows the results of the comparisons in cases 3 and 2. Regarding case 2, we can see that indeed as the dimension increases, we need exponentially many particles (quantified by N) to reach high accuracy of order 10−9 . This indicates that Algorithm 1, while being quite reliable, and sometimes the only available solver for low-dimensional problems, might need further improvement in high-dimensional settings.

This paper is available on arxiv under CC BY 4.0 DEED license.

L O A D I N G

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